Tchebotarev theorems for function fields

TL;DR

This paper establishes Tchebotarev theorems for function field extensions over various base fields, extending classical results to more general settings and exploring local-global phenomena.

Contribution

It generalizes Tchebotarev theorems to diverse base fields and infinite extensions, and investigates related local-global questions with new results and examples.

Findings

Tchebotarev theorems hold for function fields over multiple base fields.

The Galois group exponent is bounded by local specialization degrees.

Counter-examples highlight limitations of local-global principles.

Abstract

We prove Tchebotarev type theorems for function field extensions over various base fields: number fields, finite fields, p-adic fields, PAC fields, etc. The Tchebotarev conclusion - existence of appropriate cyclic residue extensions - also compares to the Hilbert specialization property. It is more local but holds in more situations and extends to infinite extensions. For a function field extension satisfying the Tchebotarev conclusion, the exponent of the Galois group is bounded by the l.c.m. of the local specialization degrees. Further local-global questions arise for which we provide answers, examples and counter-examples.